Pricing vulnerable options in a hybrid credit risk model driven by Heston-Nandi GARCH processes
aa r X i v : . [ q -f i n . P R ] J un Pricing vulnerable options in a hybrid credit riskmodel driven by Heston-Nandi GARCH processes ∗ GECHUN LIANG, XINGCHUN WANG † Abstract
This paper proposes a hybrid credit risk model, in closed form, to price vulnerable optionswith stochastic volatility. The distinctive features of the model are threefold. First, both theunderlying and the option issuer’s assets follow the Heston-Nandi GARCH model with theirconditional variance being readily estimated and implemented solely on the basis of the observ-able prices in the market. Second, the model incorporates both idiosyncratic and systematicrisks into the asset dynamics of the underlying and the option issuer, as well as the intensityprocess. Finally, the explicit pricing formula of vulnerable options enables us to undertake thecomparative statistics analysis.
Keywords : Vulnerable options; hybrid credit risk model; Heston-Nandi GARCH model; closedform formula.
JEL classification : G13
Vulnerable options refer to financial derivatives subject to default risk of the option’s issuers, andthey are widely traded in over-the-counter (OTC) markets. As of the first half of 2019, 3.9 trillion ∗ Gechun Liang is at Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Xingchun Wangis at the School of International Trade and Economics, University of International Business and Economics, Beijing100029, China. This study was supported by the National Natural Science Foundation of China (Nos. 11701084 and11671084) and Excellent Young Scholars Program in University of International Business and Economics (17YQ01). † Correspondence address, Office 416, Qiuzhen Building, University of International Business and Economics,Beijing 100029, China. Email: [email protected]; [email protected] . The bespokenature and the flexibility in terms of product design have helped OTC markets to thrive. Asopposed to exchange-traded derivatives for which products are limited in tenor, size and strikeranges, OTC derivatives facilitate tailoring of transactions to meet specific end-users’ needs. Inthis paper, we study vulnerable options with stochastic volatility in a hybrid credit risk modeldriven by GARCH processes.In order to study default risk of options, two types of models are widely used: structuralmodels and reduced-form models . Johnson and Stulz (1987) first investigate vulnerable optionsusing the structural approach, where default happens when the value of the option at maturityexceeds the value of the option issuer’s assets, resulting in the failure of the option issuer to honortheir obligation. This assumption is relaxed by Klein (1996), where the option issuer could holdother liabilities having the same priority as the option. Vast majority of research focuses on thestructural framework by taking into account of more factors such as stochastic interest rate, jumprisk, stochastic volatility, stochastic default barriers, and multiple counterparties . One attractivefeature of the structural approach is its ability to explain default events via the structural variablessuch as asset dynamics.As opposed to the structural approach, the reduced-form models are silent about why defaultshappen and, instead, the dynamics of default are exogenously given through a default rate, i.e.the default intensity. The latter approach is also called intensity approach. In contrast to thereduced-form approach for bond pricing where the payoff is a fixed income, the payoff of vulnerableoptions is random, so it is more challenging in reduced-form models to obtain an explicit pricingformula of vulnerable options. There are relatively few results in this direction. To name a few, Hulland White (1995) impose an independence assumption to obtain a closed-form pricing formula ofvulnerable options; Fard (2015) obtains a closed-form price for vulnerable options by assuming thatthe default intensity is captured by a mean-reverting Ornstein-Uhlenbeck process (so a negativeintensity is allowed); Antonelli et al. (2020) employ a correlation expansion approach to provide anapproximate evaluation of vulnerable option prices; and Wang (2017) obtains a closed-form solutionfor vulnerable options in a discrete-time GARCH framework.In this paper, we consider vulnerable options in a hybrid credit risk model. The model will Resource: BIS, OTC derivatives statistics, A partial list of the studies on this topic includes Rich (1996), Klein and Inglis (1999), Klein and Inglis (2001),Cao and Wei (2001), Hui et al. (2003), Liao and Huang (2005), Kao (2006), Liang and Ren (2007), Xu et al. (2012),Tian et al. (2014), Yang et al. (2014), Lee et al. (2016), Wang (2016), Wang et al. (2017), and Wang (2018).
In this section, we propose a hybrid credit risk model to price vulnerable options. An explicitpricing formula of vulnerable options is derived based on the change of measure technique and theexplicit expression of the joint characteristic function of underlying processes.
Let Q be a risk neutral probability measure on a filtered probability space (Ω , F , ( F t ) t ≥ , Q ).Consider a market with the systematic risk factor modelled by the market index M ( t ), whosedynamics follow the Heston-Nandi GARCH process, that is, ( ln M ( t ) = ln M ( t −
1) + r − h m ( t ) + p h m ( t ) Z m ( t ) ,h m ( t ) = w m + b m h m ( t −
1) + a m (cid:16) Z m ( t − − c m p h m ( t − (cid:17) , (2.1)where r is the continuously compounded interest rate for the time interval [ t − , t ], and Z m ( t )is a standard normal random variable. The conditional variance h m ( t ) of the log return between t − t is known from the information set at time t −
1, so it can be readily estimated andimplemented solely on the basis of the observables. In the driving noise term of h m ( t ), the constant a m determines the kurtosis of the noise, and the constant c m results in asymmetric influence of thenoise Z m ( t − h m ( t ) is a square-root diffusion process corresponding to the continuous time Hestonstochastic volatility model. On the other hand, it is clear that the discounted price of the marketindex is a martingale under Q . Indeed, we have E t − h M ( t ) i = E t − h M ( t − e r − h m ( t )+ √ h m ( t ) Z m ( t ) i = M ( t − e r E t − h e − h m ( t )+ √ h m ( t ) Z m ( t ) i = M ( t − e r , h m ( t ) is known given the information at time t − Z m ( t ) is a standard normal variable.Consider a stock in this market. Its return is affected by not only the systematic risk factorvia Z m ( t ) but also the idiosyncratic risk factor via an independent normal random variable Z s ( t ).Hence, the stock price S ( t ) under Q is driven by the process ( ln S ( t ) = ln S ( t −
1) + r − h s ( t ) + p h s ( t ) Z s ( t ) − β s h m ( t ) + β s p h m ( t ) Z m ( t ) ,h s ( t ) = w s + b s h s ( t −
1) + a s (cid:16) Z s ( t − − c s p h s ( t − (cid:17) , (2.2)where the constant β s captures the sensitivity of the stock price to systematic risk. Since h m ( t )and h s ( t ) are known given the information at time t −
1, the independence assumption between Z m ( t ) and Z s ( t ) implies that the discounted value of S ( t ) is also a martingale under Q , E t − h S ( t ) S ( t − i = E t − h e r − h s ( t )+ √ h s ( t ) Z s ( t ) − β s h m ( t )+ β s √ h m ( t ) Z m ( t ) i = e r E t − h e − h s ( t )+ √ h s ( t ) Z s ( t ) i E t − h e − β s h m ( t )+ β s √ h m ( t ) Z m ( t ) i = e r . We consider a European call option written on the stock with strike price K and maturity T , soits risk neutral price is given by E [ e − rT ( S ( T ) − K ) + ] if the option issuer does not default during thecontract period and is able to honor their obligation . Note that under the above GARCH framework,Heston and Nandi (2000) derived an explicit pricing formula for the European call option using thecharacteristic function of S ( t ) (see section 2 therein). When the options are traded in OTC markets, the holders may face the potential default risk thatthe issuers are not able to deliver the promised payoff. We model the default risk in a hybrid model.To this end, let N ( t ) be a doubly stochastic Poisson process (Cox process) with intensity Λ( t ), and τ be its first jump time which can be regarded as the arrival time of the default trigger event as inGu et al. (2014). A loss given default (LGD) will occur when the trigger event arrives, and it isgiven by a constant L . Furthermore, assume that the option issuer would recover from the triggerevent if the value of the issuer’s assets is larger than the LGD L . Hence, default occurs only whenthe trigger event occurs and the value of the issuer’s assets at the arrival time of the trigger eventfalls below the LGD L .Next, we model the option issuer’s assets V ( t ) and the Cox process’ intensity Λ( t ). Assumethat the return of the issuer’s assets is also affected by both the systematic and idiosyncratic risks5nd its dynamics follow ( ln V ( t ) = ln V ( t −
1) + r − h v ( t ) + p h v ( t ) Z v ( t ) − β v h m ( t ) + β v p h m ( t ) Z m ( t ) ,h v ( t ) = w v + b v h v ( t −
1) + a v (cid:16) Z v ( t − − c v p h v ( t − (cid:17) , (2.3)where Z v ( t ) is a standard normal variable independent of Z s ( t ) and Z m ( t ). Note that Z m ( t )captures the systematic risk, and Z s ( t ) and Z v ( t ) represent the idiosyncratic risks of the underlyingasset and the issuer’s assets, respectively. Similarly to β s in (2.2), β v captures the sensitivity of theissuer’s assets to the systematic risk. As for the intensity process Λ( t ), we assume that it is drivenby Z v ( t ) and Z m ( t ), the driving noise faced by the issuer. Specifically, the dynamics of Λ( t ) aregiven by Λ( t + 1) = w λ + b λ Λ( t ) + a λ ( Z m ( t )) + c λ ( Z v ( t )) . (2.4)All the parameters are non-negative to ensure that the intensity is non-negative.We are now in a position to present the hybrid credit risk model for the valuation of vulnerableoptions. To take account of the issuer’s default risk, we model the difference between the default-free value and the true value of the European option as follows: When j − < τ ≤ j, V ( j ) < L , i.e.the trigger event occurs between ( j − , j ] and the issuer’s asset value falls below the LGD, supposethe option holder will then only receive αV ( j ) /L proportion of the nominal payoff ( S ( T ) − K ) + at the maturity T , where the constant α ∈ [0 ,
1] represents the recovery rate and (1 − α ) V ( j )represents the deadweight costs associated with the bankruptcy. Hence, the expected value of thecredit value adjustment (i.e. the difference between the default-free value and the true value) is E (cid:20) e − rT (1 − αV ( j ) L )( S ( T ) − K ) + (cid:21) . conditional on the event { j − < τ ≤ j, V ( i ) < L } .The price of the vulnerable option at time 0 is therefore given by C = E h e − rT ( S ( T ) − K ) + i − T X j =1 E h e − rT I ( j − < τ ≤ j, V ( j ) < L )(1 − αV ( j ) L )( S ( T ) − K ) + i = E h e − rT ( S ( T ) − K ) + i − T X j =1 E h e − rT I ( j − < τ ≤ j, V ( j ) < L )( S ( T ) − K ) + i + T X j =1 E h e − rT I ( j − < τ ≤ j, V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i , (2.5)where I ( · ) is the indicator function. The first term in (2.5) is the default-free value, the secondterm represents the costs when default occurs, and the last term is the recovery value from the6efault. Note that I ( j − < τ ≤ j ) = I ( j − < τ ) − I ( j < τ ), so the last two terms in (2.5)simplify to E h I ( j − < τ ≤ j, V ( j ) < L )( S ( T ) − K ) + i = E h I ( j − < τ ≤ j ) I ( V ( j ) < L )( S ( T ) − K ) + i = − E h I ( j < τ ) I ( V ( j ) < L )( S ( T ) − K ) + i + E h I ( j − < τ ) I ( V ( j ) < L )( S ( T ) − K ) + i = − E h e − P jk =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i + E h e − P j − k =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i , and E h I ( j − < τ ≤ j, V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i = E h I ( j − < τ ≤ j ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i = − E h I ( j < τ ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i + E h I ( j − < τ ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i = − E h e − P jk =1 Λ( k ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i + E h e − P j − k =1 Λ( k ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i . In turn, we have C = e − rT (cid:16) E h ( S ( T ) − K ) + i + T X j =1 E h e − P jk =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i − T X j =1 E h e − P j − k =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i − T X j =1 E h e − P jk =1 Λ( k ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i + T X j =1 E h e − P j − k =1 Λ( k ) I ( V ( j ) < L ) αV ( j ) L ( S ( T ) − K ) + i(cid:17) . (2.6) Remark 2.1
In the proposed framework, the correlation coefficient between the underlying assetand the issuer’s assets is given by
Cov t (ln S ( t +1) S ( t ) , ln V ( t +1) V ( t ) ) q Var t (ln S ( t +1) S ( t ) ) q Var t (ln V ( t +1) V ( t ) ) = Cov t ( β s p h m ( t + 1) Z m ( t + 1) , β v p h m ( t + 1) Z m ( t + 1)) p h s ( t + 1) + β s h m ( t + 1) p h v ( t + 1) + β v h m ( t + 1)7 β s β v h m ( t + 1) p h s ( t + 1) + β s h m ( t + 1) p h v ( t + 1) + β v h m ( t + 1) . When β s = 0 or β v = 0 , the underlying asset and the issuer’s assets are not correlated with eachother. On the other hand, when h s ( t + 1) ≡ (i.e. w s = b s = a s = 0 ) and h v ( t + 1) ≡ (i.e. w v = b v = a v = 0 ), both the underlying asset and the issuer’s assets are only driven by Z m ( t ) , andthe correlation coefficient becomes to be ± . In this sense, we can view Z m ( t ) as a common riskfactor in the returns on the underlying asset and the issuer’s assets, and the issuer could hedge theoption position by directly trading the underlying asset. Thus, Z m ( t ) could represent not only thesystematic risk factor (though such an interpretation is the most typical example). In order to obtain an explicit pricing formula for vulnerable options in the proposed framework, wefirst derive the joint conditional generating function of the underlying processes. To this end, let f ( t ; φ , φ , φ , φ ) denote the conditional generating function given below, f ( t ; φ , φ , φ , φ ) = E t h exp n φ ln S ( T ) + φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) oi , where j ≤ T and 0 ≤ t ≤ T . Specially, f ( t ; φ , , ,
0) is the conditional generating function ofthe underlying asset and can be used to derive the default-free value of the European option as inHeston and Nandi (2000). In addition, f ( t ; φ , , φ , φ ) can be employed to obtain the closed-formpricing formula of vulnerable options in the reduced-form models, which is a special case of theproposed hybrid credit risk model (see section 2.4).In the proposed framework, the explicit expression of f ( t ; φ , φ , φ , φ ) is available and givenin the following proposition. Proposition 2.1
The conditional generating function has the following form f ( t ) = exp n φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) + φ ln S ( t ) + A ( t )+ A ( t ) h m ( t + 1) + A ( t ) h s ( t + 1) o , (2.7) for j ≤ t ≤ T , where A ( t ) , A ( t ) and A ( t ) ( j ≤ t ≤ T ) are defined recursively with terminalconditions A ( T ) = A ( T ) = A ( T ) = 0 by the following expressions A ( t ) = φ r + A ( t + 1) + w m A ( t + 1) + w s A ( t + 1) −
12 ln(1 − a m A ( t + 1)) For convenience, we use the more parsimonious notation f ( t ) to indicate f ( t ; φ , φ , φ , φ ), and similarly for A i ( t ) and B i ( t ).
12 ln(1 − a s A ( t + 1)) ,A ( t ) = b m A ( t + 1) − φ β s + φ β s c m − c m + ( φ β s − c m ) − a m A ( t + 1) ,A ( t ) = b s A ( t + 1) − φ + φ c s − c s + ( φ − c s ) − a s A ( t + 1) . For t < j , f ( t ) = exp n φ ln V ( t ) + φ t X k =1 Λ( k ) + φ ln S ( t ) + B ( t )+ B ( t ) h m ( t + 1) + B ( t ) h s ( t + 1) + B ( t ) h v ( t + 1) + B ( t + 1)Λ( t + 1) o , (2.8) where B k ( t ) , k = 0 , , , , ( t < j ) can be obtained recursively by the following expressions B ( t ) = B ( t + 1) + ( φ + φ ) r + w m B ( t + 1) + w s B ( t + 1) + w v B ( t + 1) + w λ B ( t + 1) −
12 ln(1 − a m B ( t + 1) + a λ B ( t + 1))) −
12 ln(1 − a s B ( t + 1))) −
12 ln(1 − a v B ( t + 1) + c λ B ( t + 1))) ,B ( t ) = b m B ( t + 1) − φ β v − φ β s + a m c m B ( t + 1)+ 2( a m c m B ( t + 1) − ( φ β v + φ β s ) / − a m B ( t + 1) + a λ B ( t + 1)) ,B ( t ) = b s B ( t + 1) − φ + a s c s B ( t + 1) + 2( a s c s B ( t + 1) − φ / − a s B ( t + 1) ,B ( t ) = b v B ( t + 1) − φ + a v c v B ( t + 1) + 2( a v c v B ( t + 1) − φ / − a v B ( t + 1) + c λ B ( t + 1)) ,B ( t ) = b λ B ( t + 1) + φ . Moreover, terminal conditions B k ( j − , k = 0 , , , , are determined by A ( j ) , A ( j ) and A ( j ) as follows: B ( j −
1) = A ( j ) + ( φ + φ ) r + w m A ( j ) + w s A ( j ) −
12 ln(1 − a m A ( j )) −
12 ln(1 − a s A ( j )) ,B ( j −
1) = b m A ( j ) − φ β v − φ β s + a m c m A ( j ) + 2( a m c m A ( j ) − ( φ β v + φ β s ) / − a m A ( j ) ,B ( j −
1) = b s A ( j ) − φ + a s c s A ( j ) + 2( a s c s A ( j ) − φ / − a s A ( j ) ,B ( j −
1) = − φ + 12 φ ,B ( j −
1) = φ . Proof. See the appendix. 9e are ready to obtain the closed form pricing formula of the vulnerable option price in (2.5).
Theorem 2.1
The price of the vulnerable European call option with strike price K and maturity T is given by C = e − rT (cid:16) f (0; 1 , , ,
0) + 1 π Z ∞ Re h e − iφ ln K f (0; 1 + iφ , , , iφ i d φ − K − Kπ Z ∞ Re h e − iφ ln K f (0; iφ , , , iφ i d φ + T X j =1 (cid:16) Π ,j − Π ,j − αL Π ,j + αL Π ,j − K (Π ,j − Π ,j − αL Π ,j + αL Π ,j ) (cid:17)(cid:17) , where Π j, - Π j, are given in (A.5)-(A.12). Proof. See the appendix.
Reduced-form models can be seen as a special case of the proposed hybrid credit risk model. Toconnect with the reduced-form model, we discard the LGD L and only check the default triggerevent τ . Hence, the price in the reduced-form model is given by C R = E h e − rT ( S ( T ) − K ) + i − (1 − α ) T X j =1 E h e − rT I ( j − < τ ≤ j )( S ( T ) − K ) + i . (2.9)The vulnerable option price in (2.9) is given in the following theorem. Theorem 2.2
In the reduced-form model, the price of the vulnerable European call option withstrike price K and maturity T is given by C R = e − rT (cid:16) f (0; 1 , , ,
0) + 1 π Z ∞ Re h e − iφ ln K f (0; 1 + iφ , , , iφ i d φ − K − Kπ Z ∞ Re h e − iφ ln K f (0; iφ , , , iφ i d φ − (1 − α ) T X j =1 ( ¯Π j, − ¯Π j, ) (cid:17) , where ¯Π j, = 12 f (0; 1 , , , −
1) + 1 π Z ∞ Re h e − iφ ln K f (0; 1 + iφ , , , − iφ i d φ − K f (0; 0 , , , − − Kπ Z ∞ Re h e − iφ ln K f (0; iφ , , , − iφ i d φ , ¯Π j, = 12 f (0; 1 , , − , −
1) + 1 π Z ∞ Re h e − iφ ln K f (0; 1 + iφ , , − , − iφ i d φ − K f (0; 0 , , − , − − Kπ Z ∞ Re h e − iφ ln K f (0; iφ , , − , − iφ i d φ . L (so we are silent about why the issuer defaults). In section 3, we willcompare the proposed hybrid model with the above reduced-form model numerically. In this section, we undertake comparative statistics analysis for the vulnerable option prices inthe proposed hybrid credit risk model. For comparison purpose, we also report the values of thecorresponding European options without default risk and vulnerable options in the reduced-formmodel in section 2.4. In particular, the default premiums, i.e. the price differences between thevanilla European options and the above two vulnerable option prices, are illustrated.In order to calculate the prices, we use the values of the parameters listed in Table 1. Theseparameter values in the dynamics of the market index and the underlying asset are also used in Suand Wang (2019), and they are estimated based on the daily closing values of the S&P 500 indexand its five largest stocks for the period from January 3, 2000 to May 31, 2018. In addition, theinitial variance values are set to be squared stationary volatilities. The parameter values in theintensity process can produce average cumulative default rates for corporate bonds with a creditrating of B, i.e., 5 . . .
89% and 34 .
47% for 1 .
0, 3 .
0, 5 . . β s and β v , and the cor-responding price difference is shown in Figure 4. Recall that β s represents the sensitivity of the11able 1: Parameter Values
Parameters in the market index dynamicsInitial price M (0) =1Initial variance h m (0) =3.27E-02Parameters governing variance processes w m = 7.10E-13 b m =7.67E-01 a m =2.99E-06 c m =2.65E+02Parameters in the underlying asset dynamicsInitial price S (0) =1Initial variance h s (0) + β s h m (0) =1.22E-01 β s =1.15Parameters governing variance processes w s =9.79E-07 b s =9.55E-01 a s =3.71E-06 c s =9.01E+01Parameters in the default intensityInitial intensity λ (0) =1.275E-06Parameters governing default intensities w λ =8.637E-07 a λ =1.372E-10 b λ =9.949E-01 c λ =1.372E-10Parameters in the value of the issuer’s assetsInitial price V (0) =1Initial variance h v (0) + β v h m (0) =1.22E-01 β v =1.15Parameters governing variance processes w v =9.79E-07 b v =9.55E-01 a v =3.71E-06 c v =9.01E+01Other parametersInterest rate r = 0 . K = 1Maturity T = 2 . α = 0 . L = 9012 .5 1 1.5 2 2.5 3 3.5 4 4.5 500.0050.010.0150.020.0250.030.0350.040.045 Maturity P r i c e D i ff e r en c e Figure 1: Option price differences against maturities. The solid line corresponds to the pricedifference between the default-free model and the proposed hybrid model, and the dot-dashed linecorresponds to the price difference between the default-free model and the reduced-form model.
Strike Prices P r i c e D i ff e r en c e Figure 2: Option price differences against strike prices. The solid line corresponds to the pricedifference between the default-free model and the proposed hybrid model, and the dot-dashed linecorresponds to the price difference between the default-free model and the reduced-form model.13 .2 0.4 0.6 0.8 1 1.20.120.130.140.150.160.170.180.190.20.21
The values of β s O p t i on P r i c e s (a) The values of β v O p t i on P r i c e s (b) Figure 3: Option prices against the values of β s and β v . The solid, dotted and dot-dashed linescorrespond to default-free option prices, option prices in the proposed hybrid model and optionprices in the reduced-form model, respectively. 14 −3 The values of β s P r i c e D i ff e r en c e (a) −3 The values of β v P r i c e D i ff e r en c e (b) Figure 4: Option price differences against the values of β s and β v . The solid line corresponds to theprice difference between the default-free model and the proposed hybrid model, and the dot-dashedline corresponds to the price difference between the default-free model and the reduced-form model.15 .3 0.35 0.4 0.45 0.5 0.55 0.600.0050.010.015 Recovery Rate P r i c e D i ff e r en c e Figure 5: Option price differences against recovery rates. The solid line corresponds to the pricedifference between the default-free model and the proposed hybrid model, and the dot-dashed linecorresponds to the price difference between the default-free model and the reduced-form model.
Caused Loss L P r i c e D i ff e r en c e Figure 6: Option price differences against different caused losses. The solid line corresponds to theprice difference between the default-free model and the proposed hybrid model, and the dot-dashedline corresponds to the price difference between the default-free model and the reduced-form model.16tock price against systematic risk. The option prices will increase with larger β s , i.e. with largersystematic risks. Intuitively, with a larger value of β s , the value of the underlying asset becomesmore volatile. Thus, it is more likely the option is in-the-money and, therefore, its price becomeshigher. On the other hand, since β v captures the sensitivity of the issuer’s assets to systematic risk,larger β v means the issuer’s assets become more risky and, as a result, the issuer is more likely todefault. Therefore, one might expect the option prices in the hybrid model become smaller withlarger β v . However, this is not the case. We observe from Figure 3(b) a higher option price withincreasing β v . This is because larger β v also means the underlying assets and the issuer’s assetsare more likely to be correlated, which in turn makes option issuers less likely to default when calloptions end in the money, yielding a higher option price consequently.Figure 5 depicts the price difference with different recovery rates. Intuitively, a higher recoveryrate corresponds to a higher option price. However, the effects of recovery rates in the hybrid modelare not as significant as those in the reduced-form model. Figure 6 shows the price difference withdifferent losses (i.e. different values of LGD). The option prices without default risk and the valuesof options in the reduced-form model are not affected by the caused losses. In the hybrid model,it is more likely that default occurs with a higher value of losses, resulting in a lower option priceand a higher default premium. In this paper, we contribute to the literature on vulnerable options by working under a hybrid creditrisk model. The proposed hybrid credit risk model incorporates the features of both structural andreduced-form models. The dynamics of the market index, as well as the dynamics of the underlyingassets and option issuer’s assets are driven by Heston-Nandi GARCH processes. The underlyingintensity process is exposed to both systematic risk and idiosyncratic risk. In this way, all thedynamics are correlated with each other through the systematic risk factor. Finally, we derive anexplicit pricing formula of vulnerable options and perform numerical analysis to illustrate optionprices.
Acknowledgement
The authors would like to thank the anonymous referee and the editor for their helpful commentsand valuable suggestions that led to several important improvements. All errors are our responsi-17ility.
Appendix
Proof of Proposition 2.1:
We first focus on the case j ≤ t ≤ T . Note that given the information at time t , V ( j ), Λ( j ) and P j − k =1 Λ( k ) are all known. Therefore, we obtain that f ( t ) = E t h exp n φ ln S ( T ) + φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) oi = e φ ln V ( j )+ φ Λ( j )+ φ P j − k =1 Λ( k ) E t h exp n φ ln S ( T ) oi . In addition, at time T , S ( T ) is also known, it follows that f ( T ) = e φ ln V ( j )+ φ Λ( j )+ φ P j − k =1 Λ( k ) E T h exp n φ ln S ( T ) oi = e φ ln S ( T )+ φ ln V ( j )+ φ Λ( j )+ φ P j − k =1 Λ( k ) , which in turn implies that A ( T ) = A ( T ) = A ( T ) = 0 . According to the law of iterated expectations, we have that E t h exp n φ ln S ( T ) oi = E t h E t +1 h exp n φ ln S ( T ) oii = E t h exp n φ ln S ( t + 1) + A ( t + 1) + A ( t + 1) h m ( t + 2) + A ( t + 1) h s ( t + 2) oi . Substituting the dynamics of ln S ( t + 1), h m ( t + 2) and h s ( t + 2) yields that E t h exp n φ ln S ( T ) oi = E t h exp n φ ln S ( t + 1) + A ( t + 1) + A ( t + 1) h m ( t + 2) + A ( t + 1) h s ( t + 2) oi = E t h exp n φ ln S ( t ) + φ r − φ h s ( t + 1) + φ p h s ( t + 1) Z s ( t + 1) − φ β s h m ( t + 1) + φ β s p h m ( t + 1) Z m ( t + 1) + A ( t + 1)+ A ( t + 1) (cid:16) w m + b m h m ( t + 1) + a m ( Z m ( t + 1) − c m p h m ( t + 1)) (cid:17) + A ( t + 1) (cid:16) w s + b s h s ( t + 1) + a s ( Z s ( t + 1) − c s p h s ( t + 1)) (cid:17)oi . Ee a ( Z + b ) = e − ln(1 − a )+ ab − a with Z being a standard normal variable andsome algebra shows that A ( t ) = φ r + A ( t + 1) + w m A ( t + 1) + w s A ( t + 1) −
12 ln(1 − a m A ( t + 1)) −
12 ln(1 − a s A ( t + 1)) ,A ( t ) = b m A ( t + 1) − φ β s + φ β s c m − c m + ( φ β s − c m ) − a m A ( t + 1) ,A ( t ) = b s A ( t + 1) − φ + φ c s − c s + ( φ − c s ) − a s A ( t + 1) . Hence, A ( t ), A ( t ) and A ( t ) ( j ≤ t ≤ T ) can be obtained recursively with terminal conditions A ( T ) = A ( T ) = A ( T ) = 0 and the above expressions.In what follows, we turn to the case t < j . Applying the law of iterated expectations to f ( t )yields that f ( t ) = E t h exp n φ ln S ( T ) + φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) oi = E t h E t +1 h exp n φ ln S ( T ) + φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) oii = E t h f ( t + 1) i = E t h exp n φ ln V ( t + 1) + φ t +1 X k =1 Λ( k ) + φ ln S ( t + 1) + B ( t + 1)+ B ( t + 1) h m ( t + 2) + B ( t + 1) h s ( t + 2) + B ( t + 1) h v ( t + 2) + B ( t + 1)Λ( t + 2) oi . Substituting the dynamics of ln V ( t + 1), Λ( t + 2), ln S ( t + 1), h m ( t + 2), h s ( t + 2) and h v ( t + 2)yields that f ( t ) = E t h exp n φ ln V ( t + 1) + φ t +1 X k =1 Λ( k ) + φ ln S ( t + 1) + B ( t + 1)+ B ( t + 1) h m ( t + 2) + B ( t + 1) h s ( t + 2) + B ( t + 1) h v ( t + 2) + B ( t + 1)Λ( t + 2) oi = E t h exp n φ ln V ( t ) + φ r − φ h v ( t + 1) + φ p h v ( t + 1) Z v ( t + 1) − φ β v h m ( t + 1) + φ β v p h m ( t + 1) Z m ( t + 1) + φ t +1 X k =1 Λ( k )+ φ ln S ( t ) + φ r − φ h s ( t + 1) + φ p h s ( t + 1) Z s ( t + 1) − φ β s h m ( t + 1) + φ β s p h m ( t + 1) Z m ( t + 1) + B ( t + 1)19 B ( t + 1) (cid:16) w m + b m h m ( t + 1) + a m ( Z m ( t + 1) − c m p h m ( t + 1)) (cid:17) + B ( t + 1) (cid:16) w s + b s h s ( t + 1) + a s ( Z s ( t + 1) − c s p h s ( t + 1)) (cid:17) + B ( t + 1) (cid:16) w v + b v h v ( t + 1) + a v ( Z v ( t + 1) − c v p h v ( t + 1)) (cid:17) + B ( t + 1) (cid:16) w λ + b λ Λ( t + 1) + a λ ( Z m ( t + 1)) + c λ ( Z v ( t + 1)) (cid:17)oi . Rearranging terms implies that f ( t ) = E t h exp n φ ln V ( t ) + φ t X k =1 Λ( k ) + φ ln S ( t ) + B ( t + 1) + ( φ + φ ) r + w m B ( t + 1) + w s B ( t + 1) + w v B ( t + 1) + w λ B ( t + 1)+( b m B ( t + 1) − φ β v − φ β s ) h m ( t + 1)+( b v B ( t + 1) − φ ) h v ( t + 1) + ( b s B ( t + 1) − φ ) h s ( t + 1)+( b λ B ( t + 1) + φ )Λ( t + 1) + Φ m + Φ s + Φ v oi , whereΦ s = φ p h s ( t + 1) Z s ( t + 1) + a s B ( t + 1)( Z s ( t + 1) − c s p h s ( t + 1)) , Φ v = φ p h v ( t + 1) Z v ( t + 1) + a v B ( t + 1)( Z v ( t + 1) − c v p h v ( t + 1)) + c λ B ( t + 1)( Z v ( t + 1)) , Φ m = ( φ β v + φ β s ) p h m ( t + 1) Z m ( t + 1) + a m B ( t + 1)( Z m ( t + 1) − c m p h m ( t + 1)) + a λ B ( t + 1)( Z m ( t + 1)) . In order to obtain the explicit expression of f ( t ), we only need to calculate E t [ e Φ m +Φ s +Φ v ] = E t [ e Φ m ] E t [ e Φ s ] E t [ e Φ v ]. Note that Φ s , Φ m and Φ v have similar forms and all can be obtained basedon the following form, E [exp { µ √ hZ + µ ( Z − µ √ h ) + µ Z } ] , where µ , µ , µ and µ are all constants and Z is a standard normal variable. Using the fact that Ee a ( Z + b ) = e − ln(1 − a )+ ab − a , we have that E [exp { µ √ hZ + µ ( Z − µ √ h ) + µ Z } ]= E [exp { ( µ + µ ) Z − µ µ − µ / Z √ h + µ µ h } ]= E [exp { ( µ + µ ) (cid:16) Z − µ µ − µ / µ + µ √ h (cid:17) − ( µ µ − µ / µ + µ h + µ µ h } ]20 e µ µ h − ( µ µ − µ / µ µ h E [exp { ( µ + µ ) (cid:16) Z − µ µ − µ / µ + µ √ h (cid:17) } ]= exp { µ µ h − ( µ µ − µ / µ + µ h −
12 ln(1 − µ + µ )) + ( µ + µ )( µ µ − µ / µ + µ ) − µ + µ ) h } = exp {−
12 ln(1 − µ + µ )) + (cid:16) µ µ − ( µ µ − µ / µ + µ + ( µ + µ )( µ µ − µ / µ + µ ) − µ + µ ) (cid:17) h } = exp {−
12 ln(1 − µ + µ )) + (cid:16) µ µ + 2( µ µ − µ / − µ + µ ) (cid:17) h } . (A.1)Therefore, we can write f ( t ) in the following form f ( t ) = exp n φ ln V ( t ) + φ t X k =1 Λ( k ) + φ ln S ( t ) + B ( t )+ B ( t ) h m ( t + 1) + B ( t ) h s ( t + 1) + B ( t ) h v ( t + 1) + B ( t + 1)Λ( t + 1) o , where B ( t ) = B ( t + 1) + ( φ + φ ) r + w m B ( t + 1) + w s B ( t + 1) + w v B ( t + 1) + w λ B ( t + 1) −
12 ln(1 − a m B ( t + 1) + a λ B ( t + 1))) −
12 ln(1 − a s B ( t + 1))) −
12 ln(1 − a v B ( t + 1) + c λ B ( t + 1))) ,B ( t ) = b m B ( t + 1) − φ β v − φ β s + a m c m B ( t + 1)+ 2( a m c m B ( t + 1) − ( φ β v + φ β s ) / − a m B ( t + 1) + a λ B ( t + 1)) ,B ( t ) = b s B ( t + 1) − φ + a s c s B ( t + 1) + 2( a s c s B ( t + 1) − φ / − a s B ( t + 1) ,B ( t ) = b v B ( t + 1) − φ + a v c v B ( t + 1) + 2( a v c v B ( t + 1) − φ / − a v B ( t + 1) + c λ B ( t + 1)) ,B ( t ) = b λ B ( t + 1) + φ . Now we need the terminal conditions of B k ( t ) , k = 0 , , , , t < j ). In other words, we need todetermine the values of B k ( j − , k = 0 , , , ,
4. Actually, we already have the expression of f ( j )from the case j ≤ t ≤ T we previously considered, f ( j ) = exp n φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) + φ ln S ( j ) + A ( j )+ A ( j ) h m ( j + 1) + A ( j ) h s ( j + 1) o . According to the law of iterated expectations, we have that f ( j −
1) = E j − h f ( j ) i E j − h exp n φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) + φ ln S ( j ) + A ( j )+ A ( j ) h m ( j + 1) + A ( j ) h s ( j + 1) oi . Substituting the dynamics of ln V ( j ), ln S ( j ), h m ( j + 1), and h s ( j + 1) and using (A.1) imply that f ( j −
1) = E j − h exp n φ ln V ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k ) + φ ln S ( j ) + A ( j )+ A ( j ) h m ( j + 1) + A ( j ) h s ( j + 1) oi = E t h exp n φ ln V ( j −
1) + φ r − φ h v ( j ) + φ p h v ( j ) Z v ( j ) − φ β v h m ( j ) + φ β v p h m ( j ) Z m ( j ) + φ Λ( j ) + φ j − X k =1 Λ( k )+ φ ln S ( j −
1) + φ r − φ h s ( j ) + φ p h s ( j ) Z s ( j ) − φ β s h m ( j ) + φ β s p h m ( j ) Z m ( j ) + A ( j )+ A ( j ) (cid:16) w m + b m h m ( j ) + a m ( Z m ( j ) − c m p h m ( j )) (cid:17) + A ( j ) (cid:16) w s + b s h s ( j ) + a s ( Z s ( j ) − c s p h s ( j )) (cid:17)oi = exp n φ ln V ( j −
1) + φ j − X k =1 Λ( k ) + φ ln S ( j −
1) + B ( j − B ( j − h m ( j ) + B ( j − h s ( j ) + B ( j − h v ( j ) + B ( j − j ) o , where B ( j −
1) = A ( j ) + ( φ + φ ) r + w m A ( j ) + w s A ( j ) −
12 ln(1 − a m A ( j )) −
12 ln(1 − a s A ( j )) ,B ( j −
1) = b m A ( j ) − φ β v − φ β s + a m c m A ( j ) + 2( a m c m A ( j ) − ( φ β v + φ β s ) / − a m A ( j ) ,B ( j −
1) = b s A ( j ) − φ + a s c s A ( j ) + 2( a s c s A ( j ) − φ / − a s A ( j ) ,B ( j −
1) = − φ + 12 φ ,B ( j −
1) = φ . This completes the proof of the proposition. ✷ Proof of Theorem 2.2:
First, we deal with the term E h ( S ( T ) − K ) + i . Recall the definition of f ( t ; φ , φ , φ , φ ) and note22hat f (0; iφ , , ,
0) is the characteristic function of ln S ( T ) under Q . From standard probabilitytheory (see, e.g., Kendall and Stuart (1977)), we can obtain the distribution function of ln S ( T ),that is, Q (ln S ( T ) ≤ x ) = 12 − π Z ∞ Re h e − iφ x f (0; iφ , , , iφ i d φ , which in turn implies that Q (ln S ( T ) ≥ x ) = 1 − Q (ln S ( T ) ≤ x )= 12 + 1 π Z ∞ Re h e − iφ x f (0; iφ , , , iφ i d φ . (A.2)The term E h ( S ( T ) − K ) + i can be derived after introducing a new probability measure Q definedby Q ( O ) = E h I ( O ) S ( T ) i E h S ( T ) i , for any event O ∈ F T . Obviously, the characteristic function of ln S ( T ) under Q is given by E Q h e iφ ln S ( T ) i = f (0; 1 + iφ , , , f (0; 1 , , , . In addition, under Q , it holds that Q (ln S ( T ) ≥ x ) = 12 + 1 π Z ∞ Re h e − iφ x f (0; 1 + iφ , , , /f (0; 1 , , , iφ i d φ . (A.3)Hence, (A.2) and (A.3) imply that E h ( S ( T ) − K ) + i = E h ( S ( T ) − K ) + i = E h ( S ( T ) − K ) I (ln S ( T ) ≥ ln K ) i = E h S ( T ) I (ln S ( T ) ≥ ln K ) i − KE h I (ln S ( T ) ≥ ln K ) i = E [ S ( T )] E Q h I (ln S ( T ) ≥ ln K ) i − KE h I (ln S ( T ) ≥ ln K ) i = E [ S ( T )] Q (ln S ( T ) ≥ ln K ) − KQ (ln S ( T ) ≥ ln K ) (cid:17) = 12 f (0; 1 , , ,
0) + 1 π Z ∞ Re h e − iφ ln K f (0; 1 + iφ , , , iφ i d φ − K − Kπ Z ∞ Re h e − iφ ln K f (0; iφ , , , iφ i d φ , (A.4)where in the last equality we used (A.2) and (A.3).Next, we focus on the term E h e − P jk =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i . We rewrite it as follows: E h e − P jk =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i E h e − P jk =1 Λ( k )+ln S ( T ) I ( V ( j ) < L, ln S ( T ) ≥ ln K ) i − KE h e − P jk =1 Λ( k ) I ( V ( j ) < L, ln S ( T ) ≥ ln K ) i = E h e − P jk =1 Λ( k )+ln S ( T ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i − KE h e − P jk =1 Λ( k ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i . In the following, we deal with the two parts in the above equality separately. To this end, we definea new probability measure Q ( O ) = E h I ( O ) e − P jk =1 Λ( k )+ln S ( T ) i E h e − P jk =1 Λ( k )+ln S ( T ) i , for any event O ∈ F T . Under Q , we have the joint characteristic function of − ln V ( j ) and ln S ( T )as follows: E Q h e iφ ( − ln V ( j ))+ iφ ln S ( T ) i = E h e − P jk =1 Λ( k )+ln S ( T ) E h e − P jk =1 Λ( k )+ln S ( T ) i e iφ ( − ln V ( j ))+ iφ ln S ( T ) i = 1 E h e − P jk =1 Λ( k )+ln S ( T ) i E h e ( iφ +1) ln S ( T ) − iφ ln V ( j ) − P jk =1 Λ( k ) i = f (0; iφ + 1 , − iφ , − , − f (0; 1 , , − , − . By inverting the characteristic function, we have thatΠ j, := E h e − P jk =1 Λ( k )+ln S ( T ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = E h e − P jk =1 Λ( k )+ln S ( T ) i E Q h I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = f (0; 1 , , − , − Q (cid:16) − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K (cid:17) = 14 f (0; 1 , , − , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ + 1 , , − , − iφ i d φ + 12 π Z ∞ Re h e iφ ln L f (0; 1 , − iφ , − , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ + 1 , − iφ , − , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ + 1 , iφ , − , − φ φ i(cid:17) d φ d φ . (A.5)Likewise, we work under ¯ Q defined by¯ Q ( O ) = E h I ( O ) e − P jk =1 Λ( k )+ln S ( T ) i E h e − P jk =1 Λ( k )+ln S ( T ) i , O ∈ F T , and obtain thatΠ j, := E h e − P jk =1 Λ( k ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = f (0; 0 , , − , −
1) ¯ Q (cid:16) − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K (cid:17) = 14 f (0; 0 , , − , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ , , − , − iφ i d φ + 12 π Z ∞ Re h e iφ ln L f (0; 0 , − iφ , − , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ , − iφ , − , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ , iφ , − , − φ φ i(cid:17) d φ d φ . (A.6)Hence, it holds that E h e − P jk =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i = E h e − P jk =1 Λ( k )+ln S ( T ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i − KE h e − P jk =1 Λ( k ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = Π j, − K Π j, . Similarly, E h e − P j − k =1 Λ( k ) I ( V ( j ) < L )( S ( T ) − K ) + i = E h e − P j − k =1 Λ( k )+ln S ( T ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i − KE h e − P j − k =1 Λ( k ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = Π j, − K Π j, , where Π j, := 14 f (0; 1 , , , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ + 1 , , , − iφ i d φ + 12 π Z ∞ Re h e iφ ln L f (0; 1 , − iφ , , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ + 1 , − iφ , , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ + 1 , iφ , , − φ φ i(cid:17) d φ d φ , (A.7)and Π j, := 14 f (0; 0 , , , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ , , , − iφ i d φ
25 12 π Z ∞ Re h e iφ ln L f (0; 0 , − iφ , , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ , − iφ , , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ , iφ , , − φ φ i(cid:17) d φ d φ . (A.8)Note that Π j, and Π j, have similar forms as Π j, and Π j, , and they can be obtained by replacing f (0; · , · , − , · ) in Π j, and Π j, with f (0; · , · , , · ), respectively.We next calculate E h e − P jk =1 Λ( k ) I ( V ( j ) < L ) V ( j )( S ( T ) − K ) + i . Note that E h e − P j − k =1 Λ( k ) I ( V ( j )
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ + 1 , , − , − iφ i d φ + 12 π Z ∞ Re h e iφ ln L f (0; 1 , − iφ + 1 , − , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ + 1 , − iφ + 1 , − , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ + 1 , iφ + 1 , − , − φ φ i(cid:17) d φ d φ , (A.9)and Π j, := E h e − P jk =1 Λ( k ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = E h e − P jk =1 Λ( k )+ln V ( j ) i E ¯ Q h I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = 14 f (0; 0 , , − , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ , , − , − iφ i d φ
26 12 π Z ∞ Re h e iφ ln L f (0; 0 , − iφ + 1 , − , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ , − iφ + 1 , − , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ , iφ + 1 , − , − φ φ i(cid:17) d φ d φ , (A.10)where ¯ Q ( O ) is defined by ¯ Q ( O ) = E h I ( O ) e − P jk =1 Λ( k )+ln V ( j ) i E h e − P jk =1 Λ( k )+ln V ( j ) i , for any event O ∈ F T . Therefore, we have that E h e − P jk =1 Λ( k ) I ( V ( j ) < L ) V ( j )( S ( T ) − K ) + i = E h e − P jk =1 Λ( k )+ln S ( T ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i − KE h e − P jk =1 Λ( k ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = Π j, − K Π j, . Finally, we calculate E h e − P j − k =1 Λ( k ) I ( V ( j ) < L ) V ( j )( S ( T ) − K ) + i under Q and ¯ Q defined by Q ( O ) = E h I ( O ) e − P j − k =1 Λ( k )+ln S ( T )+ln V ( j ) i E h e − P jk =1 Λ( k )+ln S ( T )+ln V ( j ) i , ¯ Q ( O ) = E h I ( O ) e − P j − k =1 Λ( k )+ln V ( j ) i E h e − P jk =1 Λ( k )+ln V ( j ) i , for any event O ∈ F T . Following along similar arguments we obtain E h e − P j − k =1 Λ( k ) I ( V ( j ) < L ) V ( j )( S ( T ) − K ) + i = E h e − P j − k =1 Λ( k )+ln S ( T ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i − KE h e − P j − k =1 Λ( k ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = Π j, − K Π j, , where Π j, := E h e − P j − k =1 Λ( k )+ln S ( T ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = 14 f (0; 1 , , , −
1) + 12 π Z ∞ Re h e − iφ ln K f (0; iφ + 1 , , , − iφ i d φ
27 12 π Z ∞ Re h e iφ ln L f (0; 1 , − iφ + 1 , , − iφ i d φ − π Z ∞ Z ∞ (cid:16) Re h e − iφ ln K + iφ ln L f (0; iφ + 1 , − iφ + 1 , , − φ φ i − Re h e − iφ ln K − iφ ln L f (0; iφ + 1 , iφ + 1 , , − φ φ i(cid:17) d φ d φ . (A.11)and Π j, := E h e − P j − k =1 Λ( k ) V ( j ) I ( − ln V ( j ) > − ln L, ln S ( T ) ≥ ln K ) i = 14 f (0; 0 , , , −